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Biot-Savart Law

Biot-Savart Law in Grade 12 Physics

Biot-Savart Law

The Biot-Savart Law is an equation describing the magnetic field generated by a constant electric current. It relates the magnetic field to the magnitude, direction, length, and proximity of the electric current. Biot–Savart law is consistent with both Ampere’s circuital law and Gauss’s theorem. The Biot Savart law is fundamental to magnetostatics, playing a role similar to that of Coulomb’s law in electrostatics.

biot- savart law

Biot Savart law was created by two French physicists, Jean Baptiste Biot and Felix Savart derived the mathematical expression for magnetic flux density at a point due to a nearby current-carrying conductor, in 1820. Viewing the deflection of a magnetic compass needle, these two scientists concluded that any current element projects a magnetic field into the space around it.
 

Through observations and calculations, they had derived a mathematical expression, which shows, the magnetic flux density of which dB, is directly proportional to the length of the element dl, the current I, the sine of the angle, and θ between the direction of the current and the vector joining a given point of the field and the current element and is inversely proportional to the square of the distance of the given point from the current element, r.

Biot Savart Law Statement

This is Biot Savart law statement:

biot savart law


Where, k is a constant, depending upon the magnetic properties of the medium and system of the units employed. In the SI system of the unit,

in Grade 12 Physics

Therefore, the final Biot Savart law derivation is,

biot savart law

Let us consider a long wire carrying a current I and also consider a point p in the space. The wire is presented in the picture below, in red color. Let us also consider an infinitely small length of the wire dl at a distance r from the point P as shown. Here, r is a distance-vector that makes an angle θ with the direction of current in the infinitesimal portion of the wire.

If you try to visualize the condition, you can easily understand the magnetic field density at point P due to that the infinitesimal length dl of the wire is directly proportional to the current carried by this portion of the wire.

As the current through that infinitesimal length of wire is the same as the current carried by the whole wire itself, we can write,

biot savart law
It is also very natural to think that the magnetic field density at that point P due to that infinitesimal length dl of wire is inversely proportional to the square of the straight distance from point P to the center of dl. Mathematically we can write this as,

biot savart law

Magnetic field
Lastly, magnetic field density at that point P due to that infinitesimal portion of the wire is also directly proportional to the actual length of the infinitesimal length dl of wire. As θ be the angle between distance vector r and direction of current through this infinitesimal portion of the wire, the component of dl directly facing perpendicular to the point P is dlsinθ Biot-Savart Law

biot savart law
Now, combining these three statements, we can write,
Magnetic field

This is the basic form of Biot Savart’s Law

Now, putting the value of constant k (which we have already introduced at the beginning of this article) in the above expression, we get
Magnetic field

Here, μ0 used in the expression of constant k is absolute permeability of air or vacuum and its value is 4π10-7 Wb/ A-m in SI system of units. μr of the expression of constant k is the relative permeability of the medium.

Now, flux density(B) at the point P due to the total length of the current-carrying conductor or wire can be represented as,

Magnetic field

Magnetic fieldIf D is the perpendicular distance of the point P from the wire, then
Biot-Savart Law
Now, the expression of flux density B at point P can be rewritten as,

Magnetic field

As per the figure above,

Biot-Savart Law

Finally, the expression of B comes as,

Biot-Savart Law
This angle θ depends upon the length of the wire and the position of the point P. Say for a certain limited length of the wire, angle θ as indicated in the figure above varies from θ1 to θ2. Hence, magnetic flux density at point P due to the total length of the conductor is,

Biot-Savart Law

Let’s imagine the wire is infinitely long, then θ will vary from 0 to π that is θ1 = 0 to θ2 = π. Putting these two values in the above final expression of Biot Savart law, we get,

Magnetic field

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